I have a phasor sum
$a e^{j \theta} = \frac{1}{\sqrt{N}} \sum_{k=1}^{N} \alpha_k e^{j \phi_k }$
where $\phi_k = [-\pi, \pi]$, the standard deviation $\sigma_{\phi}$ of the phase is known and the mean of the phase is $\mu_\phi = 0$.
What is the standard deviation of $\theta$?
In general is very unlikely to get closed form expression in $\theta$ and this is why:
Variance of the left side can be calculated by using Taylor expression for the exponent. Doing this we have to engage not only mean and variance of $\theta$ but all higher moments. Higher moments expression can be presented in closed form for only certain probability distribution functions. Then we get an equation on $\theta$ that, in general case, can not be solved analytically.
Right side can be calculated knowing probability distribution function for $\phi_i$ and its higher moments. If all $\phi_i$ are independent and means are zero - it simplify the calculation, but knowledge of all higher moments or probability distribution function is necessary.
If your variables come from certain known physical process where distribution function is known - it still require some effort to plug everything in and put some effort to come with closed form solution.